Quadratic equation calculator
Enter a, b and c for ax² + bx + c = 0. The calculator works out the discriminant, says what kind of roots there are, then solves with the quadratic formula and writes out every step.
How it works
- Write the equation as ax² + bx + c = 0 and read off a, b and c (watch the signs).
- Work out the discriminant Δ = b² − 4ac: Δ > 0 gives two distinct real roots, Δ = 0 two equal real roots and Δ < 0 no real roots.
- Substitute into the quadratic formula x = (−b ± √Δ) / 2a.
- Simplify the surd, or give the answer correct to 3 significant figures if the question asks.
- a = 2, b = −3, c = −5
- Δ = (−3)² − 4(2)(−5) = 9 + 40 = 49 > 0, so there are two distinct real roots.
- x = (3 ± 7) / 4
To learn it properly, read Quadratic equations in one unknown: key points, steps and worked examples.
Frequently asked questions
When should I use the formula instead of factorising?
If the equation factorises easily, factorising is quicker. If you cannot find the factors, or the question asks for approximate answers, use the quadratic formula. It works for every quadratic equation.
What should I watch for when b is negative?
Use brackets when you substitute. With b = −3, b² = (−3)² = 9 and −b = 3. Many mistakes come from leaving out the brackets.
What if Δ is less than 0?
The equation has no real roots. The non-foundation part of the syllabus writes the roots as complex numbers, and the calculator shows those in the steps too.